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ON 

A NEW METHOD 



OF OBTAINING THE 



Differentials of Functions 

WITH ESPECIAL REFERENCE TO THE 

NEWTONIAN CONCEPTION OF RATES 
OR VELOCITIES 



J. MINOT RICE 

PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY 
AND 

W. 1 WOOLSEY JOHNSON 

PROFESSOR OF MATHEMATICS IN SAINT JOHN'S COLLEGE ANNAPOLIS MARYLAND 



REVISED EDITION 



NEW YORK 

D. VAN NOSTRAND PUBLISHER 
88 Murray Street and 27 Warren Street 

ist:. 



COPYRIGHT, 1875. D. VAN NOSTRAND. 



PREFACE. 



This pamphlet is a revised edition of a paper which was read 
before the American Academy of Arts and Sciences, January 
14, 1873, and subsequently published in its Proceedings. 

It is now reproduced for the purpose — first, of presenting a 
new method of deriving the Differentials of Functions by means 
of their Algebraic characteristics with the aid of a few element- 
ary properties easily established ; and, secondly, of showing that 
the method of rates or fluxions may be advantageously used 
for the purposes of instruction, and the use of infinitesimals, 
limits and series entirely avoided until the student is well 
grounded in the elements of the Calculus ; thus securing the 
advantages afforded by the real and precise definitions of this 
method, instead of sacrificing them by employing the difficult 
and readily misconceived notions of limits or infinitesimals in 
deducing the formulas for differentiation. 

In the original paper it was shown that the new method of 
deducing the differentials was applicable to all the functions 
of a single variable ; some of these applications are now 
omitted to be replaced by other methods which we consider 
preferable for the purposes of elementary instruction. 

It is our intention to publish a text-book prepared in ac- 
cordance with this plan, the first part of which has already 
been printed for the use of the cadets at the U. S. Naval 
Academy. 

J. M. E. 

Annapolis, Md., July, 1875. "W. TV. J. 



CONTENTS 



Pages. Art. 
The Newtonian Method of Fluxions 7-12 

The methods of treating the Differential Calculus in 

general use at the present time 1-2 

Extract from Todhunter's Differential Calculus 3 

Extract from Lacroix's Treatise on the Differential and 

Integral Calculus 6 

Extract from Carnot's Metaphysics of the Infinitesimal 

Calculus r 7 

Extract from Cournot's Infinitesimal Calculus 7 

Proposed Method of Treating the Differen- 
tial Calculus 13 32 

Definitions and Notation 8-9 

Elementary Propositions 10 

Constant Ratio of the Rates of a Variable and a linear Func- 
tion 11 

Variable Ratio of the Rates of a Variable and a Func- 
tion not linear 12 

The general definition of a tangent line to a curve 12 

The Ratio of the Rates is independent of their absolute 

values 13 

Algebraic Functions 18-20 

The Differential of the Square 15-16 

The Differential of the Product 17 



6 CONTENTS. 

Pages. Art. 

Transcendental Functions 22-30 

The Differential of the Logarithmic Function 18 

The Rate or Differential of an Area 20 

Hyperbolic Areas, and Napierian Logarithms 21 

The Napierian base intermediate in value between 2 and 3 22 

The Differentials of the Trigonometric Functions 23-24 

Concluding Remarks 25-27 



THE NEWTONIM METHOD OF FLUXIONS. 



1. It is a well-known fact that writers on the Differential 
Calculus deduce the same elementary theorems from funda- 
mental conceptions which have little or no apparent resem- 
blance, and frequently employ methods which present to the 
conscientious student difficulties of a character too formidable 
to be ignored. 

Notwithstanding the unusual attention which many of the 
ablest mathematical writers since the time of Newton and 
Leibnitz have bestowed upon this subject it is undoubtedly 
true that many instructors still tacitly permit their students to 
follow the trite precept of D'Alembert, — u Allez en avant, et la 
foi vous viendra" Few habits are more pernicious to the 
student of mathematics than that of following rules founded 
upon principles which he does not thoroughly comprehend ; 
yet it is precisely this habit that D'Alembert's precept tends 
to confirm. Faith comes — only too soon. 

In the words of Condillac — u Ce n'est point par la routine 
qu'on s'instruit, c'est par sa propre reflexion ; et il est essential 
de contractor l'habitude de se rendre raison de ce qu'on fait ; 
cette habitude s'acquiert plus facilement qu'on ne pense ; et 
une fois acquise, elle ne so perd plus." 

. 2. Most of the French writers on the Calculus have adopted 
a method of treating the subject which may be characterized 



8 THE NEWTONIAN METHOD OF FLUXIONS. 

as a combination of the method of limits with that of infinites- 
imals. One of the best examples is the work of J. A. Serret, 
Paris, 1868. This method, although excellent in extensive 
treatises like those of Serret and Bertrand, seems to us far too 
difficult for a beginner not possessed of unusual mathematical 
ability, especially as it involves several fundamental proposi- 
tions very hard to comprehend. In nearly all cases, it will be 
found best for the student not to attempt these works until he 
has prepared himself by studying one of a less formidable 
character. 

3. A distinguished English writer of mathematical text- 
books (Mr. I. Todhunter), who Las himself adopted the method 
of limits, remarks that — " A difficulty of a more serious kiud 
which is connected with the notion of a limit, appears to em- 
barrass many students of this subject, namely, a suspicion that 
the methods employed are only approximative, and therefore 
a doubt as to whether the results are absolutely true. This ob- 
jection is certainly very natural, but at the same time by no 
means easy to meet, on account of the inability of the reader 
to point out any definite place at which his uncertainty com- 
mences." * This remark seems to us not only to go to the root 
of the difficulty, but also to suggest an excellent mode of testing 
the student's comprehension of the subject when taught by 
the ordinary methods. 

* Todhuniefs Differential Calculus, Macmillan and Co. London, 1860, 
page 12. 



THE NEWTONIAN METHOD OF FLUXIONS. 9 

4. Our plan is to return to the method of fluxions, and 
making use of the precise and easily comprehended definitions 
of Newton, to deduce the formulas of the Differential Calculus 
by a method which is not open to the objections which were 
largely instrumental in causing this view of the subject to be 
abandoned ; and one which is not in the slightest degree sug- 
gestive of approximation. 

Inasmuch as the method of fluxions is now nearly obsolete, 
we would refer the reader, who may desire a brief and lucid 
statement of its principles and peculiarities, to Montuclcfs 
History of Mathematics, Volume II, pages 320, 321, 322, and 
323 (the edition before us is that of 1758). 

5. The objections hitherto urged against this view of the 
subject may be divided into two classes ; first, those which refer 
to the mode of deducing the formulas, and secondly, objec- 
tions to the definitions employed. The following extract from 
Lacroix's treatise will serve as an example of the first class of 
objections. 

6. " Newton supposa les lignes engendrees par le mouve- 
ment d'un point, et les surfaces par colui d'une ligne ; et il 
appela fluxions les vitesses qui reglaient cos mouvemens. Ces 
notions, quoique tres-rigoureuses, sont e'trangeros a la Geome- 
tric, et leur application peut etre difficile. II est bien vrai 
qu'en imaginant un point qui so meuve sur une liguo, pendant 
qu'elle est omport<'o parallclonient a elle-memo, avee une vitosse 
uniforme, on peut representor une courbo quelconquo; mais la 



10 THE NEWTONIAN METHOD OF FLUXIONS. 

vitesse du point de'crivant e'tant variable a chaque instant, on 
ne peut la determiner qiCen recourant soit d la methode des 
Anciens on d* exhaustion, soit a celle des premieres et demieres 
raisons, et c'est presque toujours de celle-ci que Newton s'est 
servi ; ensorte que les fluxions n'etaient, a proprement parler 
pour lui, qu'un moyen de donner un object sensible aux quan- 
tite's sur lesquelles il operait."* 

The italics in the above extract are ours. It will be ob- 
served that the clause thus distinguished contains the point of 
M. Lacroix's objection; an examination of the following pages 
will show that it does not apply to our method. 

7. The two following extracts contain substantially all the 
objections of the second class that we have thus far met with, 
and also replies to them which seem to us to be sufficiently 
conclusive. 

f " Cette observation a ete le pretexte d'une objection elevee 
contre la methode des fluxions ; car, a-t-on dit, c'est introduire 
dans la Geometrie qui appartient aux Mathe'matiques pures, la 
notion des vitesses qui n'appartient qu'aux Mathe'matiques 
mixtes, et definir une idee qui doit etre simple, par une autre 
qui est complexe. 

* Traite du Calcul Differ entiel et du Calcul Integral. Par S. F. 
Lacroix. Seconde Edition. Tome premier. Paris, 1810. Preface, 
pages xv et xvi. 

f Garnot— Reflexions sur la Metapliysique du Calcul Infinitesimal. 
Quatrieme edition. Paris, 1860. Pages 114 et 115. 



THE NEWTONIAN METHOD OF FLUXIONS. 11 

" Mais cette objection est assez frivole : car la veritable chose 
a conside'rer est de savoir si la theorie est plus facile a saisir de 
cette maniere que d'une autre. Le classement que nous faisons 
des sciences est assez arbitraire. Nous pla^ons la Geometrie 
avant la Mecanique dans l'ordre de la sirnplicite, mais les par- 
ties transcendantes de la premiere sont bien plus abstraites que 
les parties elementaires de la seconde, et, ccnime le dit La- 
grange, fi chacun a ou croit avoir une ide'e nette de la vitesse ' ; 
ce n'est done pas prendre une marclie contraire a l'esprit des 
Mathematiques, que de definir les fluxions par les vitesses." 

* " On a reproche a Newton de faire intervenir, sans ne'ees* 
site, dans ce mode d'exposition, la notion du temps et celle du 
mouvement. Le reproche peut etre fonde, quant a la notion 
du mouvement, a laquelle rien n'oblige, en effet, de recourir ; 
mais on devait remarquer que la notion du temps intervient ici 
par la nature des choses, en raison de ce que le temps est la 
seule variable essentiellement independante, et la seule dont la 
variation soit essentiellement uniforme, ou la fluxion con- 
stante. 

" Dans tous les cas, la conception de Newton, appliquee aux 
grandeurs qui varient effectivement avec le temps, a Favantage 
de fixer la signification reeUe des fonctions derive'es, et par la 
nieme de donner a l'avance la raison du role qu'elles jouent 
dans les applications de l'analyse a la discussion des pheno- 
menes physiques. Newton se proposait aussi de fonder la theorie 

* Cournot—TJuorie des Fonctions et da Calcul Infinitesimal Paris, 
1841. 



12 THE NEWTONIAN METHOD OF FLUXIONS. 

des fonctions sur une idee que l'espiit put saisir directement, 
sans passer par la consideration des limites et sans s'assujettir 
a une march e jusqu'a un certain point detournee et indirecte. 
II entendait exprimer directement la continuity dans la varia- 
tion des grandeurs, au nioyen du phenomena le plus familier 
ou cette continuite tombe sous les sens. On a objecte, d'apies 
d'Alernbert, que, pour definir une vitesse continuellement 
variable, il faut toujours recourir a la consideration des lim- 
ites ; mais, en faisant cette objection, on a mal a propos sub- 
crdonne la precision des iiees a leur definition logique. Un 
concept existe dans l'entendement, indtpendamment de la 
definition qu ? on en donne ; et souvent L'idee la plus simple dans 
Tentendementne comporte qu'une definition corapliquee, quand 
elle n'echappe pas a toute definition. Tout le monde a une 
idee directe et exacte de la similitude de deux corps, quoique 
peu de gens puissent entendre les definitions compliquees que 
les ge'ometres ont donnees de la similitude."' 

The reader is desired to notice particularly the positive 
advantages of the Newtonian method which M. Cournot has so 
clearly and forcibly set forth in the second paragraph of the 
above extract. 



PROPOSED METHOD OF TREATING THE 
DIFFERENTIAL CALCULUS. 



DEFINITIONS AND NOTATION. 

8. When a quantity varies uniformly, the constant numeri- 
cal measure of its rate is the increment received in the unit of 
time. When, however, the variation is not uniform, we would 
define the numerical measure of the rate at any instant as the 
increment which would he received in a unit of time, if the 
rate remained uniform from and after the given instant. 

This definition corresponds with the usage of Mechanics, in 
accordance with which a body moving with a variable velocity 
is said to have at a given instant a velocity which would carry 
it thirty-two feet in one second. 

9. To avoid departing too much from well-established usage, 
the term differential will be frequently used in this paper in- 
stead of rate. 

The rate or differential of x will be denoted by dx, and that 
of f(x) by d[f(x)] (these symbols being always used by us 
to denote finite quantities.) 

The rate of the independent variable, or the value of dx, is 
regarded as arbitrary in the same sense in which the value of 
the variable x is itself arbitrary. 



14 PROPOSED METHOD OF TREATING 

Thus, particular values of these two quantities may consti- 
tute the data of a question like the following : What is the 
value of d(x 2 ) when x has the value 10, and dx the value 4 ? 

To differentiate a function of x is to express d[f(x)] in 
terms of x and dx in such a manner as to furnish a general 
formula by which d[f(x)'] may be computed for any given 
values of x and of dx. 

ELEMENTARY PROPOSITIONS. 

10. The following propositions are immediate deductions 
from the above method of measuring rates : — 

I. The Differential of 'x 4- h. 

Since any simultaneous increments of x and of x + h must 
be identical, the increments which would be received by each, 
if they continued to vary uniformly with the rates denoted by 
dx and d(x + A), are equal. Hence the rates are equal, or 

d(x + h) = dx. 

11. The Differential ofx + y. 

Since any increment of x + y is the sum of the simultaneous 
increments of x and of y, the same relation exists between the 
increments which would be received if x and y (and conse- 
quently x + y) continued to vary uniformly with the rates de- 
noted by dx, dy, and d(x 4- y). Hence 

d(x + y) = dx + dy. 
III. The Differential of 'mx, 



THE DIFFERENTIAL CALCULUS. 15 

Since any increment of mx must be m times the correspond- 
ing increment of x, the same relation must exist between the 
increments which would be received if x (and consequently rax) 
continued to vary uniformly with the rates denoted by dx and 
d(wx). Therefore 

d(mx) — mdx. 

THE RATIO OF THE RATES OF A VARIABLE AND ITS 
FUNCTION. 

11. Let y denote a linear function of x, that is 

y = mx + b (1) 

By propositions I and III, 

dy = mdx, 

or dy 



dx 



=m (2) 



In this case, the ratio of the rate or differential of the func- 
tion to that of the independent variable is constant, its value 
being independent, not only of x ) but also of dx. Thus, if we 
give to dx any arbitrary value, it is evident from equation (2), 
that dy must take a corresponding value such that the ratio 
of these quantities shall always retain the constant value m. 
Assuming rectangular co-ordinate axes, if y be made the ordi- 
nate corresponding to X as an abscissa, the point (x } y) will, as 
x varies, generate a straight line. Tho direction of the 
motion of the point is constant, and depends upon the value of 
m. Since - is equal to >/i, it is the trigonometrical tangent 



16 PROPOSED METHOD OP TREATING 

of the constant inclination of the direction of the generating 
point to the axis of x. 

When y is not a linear function of x, the direction of the 
motion of the generating point is variable, and consequently 

the value of -7- is variable. 
dx 

12. Making now the arbitrary quantity dx a constant, dy 
will be a variable. Suppose that, the generatrix having 
arrived at a given point, the ordinate y continues to vary uni- 
formly with the rate denoted by dy at the given point ; the 

dy 
value of — will then become constant. The generatrix will 

now continue to move uniformly in the direction of the curve at the 

dt/ 
given point, and therefore the value which -^ has at this point 

is that of the trigonometrical tangent of the inclination of the 
curve to the axis of x at this point. The line now described 
by the generatrix is called a tangent line to the curve, in ac- 
cordance with the following general definition : The tangent 
line to a curve at a give?i point is the line passiyig through the 
point, and having the direction of the curve at that point. 

IV. The Ratio of the Rates is independent of their absolute 
values. 

13. Since the direction of the curve (or of the tangent line) 
at the point having a given abscissa is determined by the form 
of the fanction, or equation to the curve, the value of — ^, 
which is the trigonometrical tangent of the inclination of this 
direotion, must be independent of the arbitrary quantity dx, 
which merely determines the velocity of the generating point. 



THE DIFFERENTIAL CALCULUS. 17 

In general, the value of ■ ^V* -* will change with that of x; 
— *~ is, therefore, independent of dx, but is generally a 
funotion of x. 

c7[/(a*)], when expressed in terms of x and dx, is of the 
form 

*[/(*)]-/(*).*■ 

in which f'{x) is another function of x. 

In the ordinary methods the introduction of an equivalent 
proposition is, for the most part, avoided, by rejecting from the 
ultimate value of A [/(#)] all terms containing powers of Ax 
higher than" the first. 

14. "We shall now proceed to show how, from the four ele- 
mentary propositions, the differentials of the functions both 
algebraic and transcendental may be deduced. These prop- 
ositions are here recapitulated for convenience of reference : — 

I. d(x+h)=dx. 

II. d(x+y)=zdx + dy. 

III. d(mx)—mdx. 

IV. =f'(x), a function of x } independent of dx. 
(X x 



18 PROPOSED METHOD OF TREATING 



ALGEBRAIC FUNCTIONS. 

THE DIFFERENTIAL OF THE SQUARE. 

15. In establishing the formulas for the differentiation of the 
simple algebraic functions of an independent variable, we find 
it most convenient to begin with the square. 

"We first deduce a relation between two values of the deriva- 
tive of the function, and the corresponding values of the inde- 
pendent variable ; for this purpose we assume two values of the 
variable having a constant ratio m, thus : 

z=mx y then will dz=mdx (1) 

and 2 9 =wV, .\ d{z i )=m , d{x) (2) 

Dividing equation (2) by equation (1), 

d ^l=m^l, (3) 

uz dx v ' 



a relation between two values of the derivative ; putting for m 

d(z*) __ z d{x*) ( 

dz x ' dx ' ^ ' 



its value — we obtain 

x 



the relation required. Dividing by z to separate the variables, 

we have 

I 3*11 = 1 ^1 (5) 

z ' dz x' dx ' 

This equation is true for all values of the quantities x, z, dx 

d(z z ) d(x 2 ) 

and dz ; for, by Theorem IV, — r-^is independent oi^dz, and— — 

dz dx 

is independent of dx } the derivatives being functions respec- 



THE DIFFERENTIAL CALCULUS, 19 

tively of z and of x simply ; moreover, the equality exists in- 
dependently of any particular value of m ) since it has been elimi- 
nated. 

The first member of this equation, being independent of the 
value of dzy can be a function of no variable quantity except z, 
and so likewise the second member can be a function of no 
variable quantity except x. 

If, therefore, we denote x 2 by/(#), and adopt the usual no- 
tation for the derivative, we may write equation (5) thus : — 

r/M =!•/(*) (&') 

Now, since the form of the first member of this equation is 
the same as that of the second member, it follows that the 

1 , 1 d f x*) 
value of the expression ~'f(x) or — — does not change 

when x is changed to z } the latter representing, it must be re- 
membered, any other value of the independent variable ; the 
value of this expression is therefore constant, and denoting it 

by 0, we write 

1 d(x*) 

whence 

d(x*)=cxd.r (7) 

To determine the unknown constant c we apply this result to 
the identity 

(*+A)«=*»+ 2Ax+A 1 ; (8) 

differentiating each member by formula (7), we have 

c{x+h)d{x + h)=cxdx+'lhdx \ (9) 



20 PROPOSED METHOD OF TREATING 

since d(x-\-h)=zdx, equation (9) reduces to 

chdx=L2hdx, 
or 

(c-2)hdz=z0; 
and, since h and dx are arbitrary quantities, we have 

c=2, 
which substituted in (7) gives 

dix*) = 2xdx. 

16. The process of which the above is an example may be 
applied in the case of all those fuuctions whose differentials it 
is desired to deduce independently. Although it is somewhat 
elaborately explained in the above application, we add the fol- 
lowing more general description of the method pursued. 

We assume a new variable z, connected with x by a relation 
admitting of a comparison of dz and dx> and at the same time 
such that d[f{z)1 and d[f(x)] may likewise be compared ; 
in other words, such that the relation between z and x> and also 
that between f(z) and/(j) can be differentiated without the 
introduction of unknown differentials except those denoted by 
d[f(z)]&n&d[f(x)]. 

By division, the ratios — jg — aad — ^—- are introduced in 

a single equation. The arbitrary constant introduced in the 
assumed relation between z and x is then eliminated, and the 
equation reduced to such a form that one member is apparently 
a function of z, and the other of x. This last process we call 
the separation of the variables. 



THE DIFFERENTIAL CALCULUS. 21 

As x and z may denote any two values of the independent 
variable, the apparent functions mentioned above will neces- 
sarily be identical in form, and (since they constitute the two 
members of an equation) identical also in value. This value 
will be constant, since either member of the equation is a 
functional expression, which does not change its value with x. 

The determination of this constant is then effected by the 
differentiation of some algebraio identity. 

17. The following method of deducing the differential of the 
product from that of the square is substantially the same as 
the one employed in Vince's Fluxions. 

THE DIFFERENTIAL OF THE PRODUCT. 

If x and y denote any two variables, xy is a function of 
both, and its differential depends upon x } y y dx, and dy. 

In order to derive this differential, we express xy by means 
of squares, since we have already obtained a formula for the 
differentiation of the square. Thus, from the identity 

(x + y)*=x* + 2xy + y*, 
we derive xy = $(x + yf - \t? — \y*. 

Differentiating, d{xy) = (x + y) (dx -h dy) — xdx —ydy t 
or d( x \j) = yd& ~r xdy. 

From the differential of the product, that of the quotient and 
power are readily deduced by the ordinary methods. We pass 
now to Transcendental Functions. 



22 PROPOSED METHOD OF TREATING 

TRANSCENDENTAL FUNCTIONS. 

THE DIFFERENTIAL OF THE LOGARITHMIC FUNCTION. 

18. To deduce the differential of the logarithmic function, 
we employ the method exemplified in the case of the square. 

The symbol log x is here used to denote the logarithm of x to 
any base, and log b # is used when we wish to designate a particu- 
lar base b. 

Let z === mx, then will dz = mdx, (1) 

logz = logm + logs, .-. d(\ogz) = d(\ogx) (2) 

Dividing (2) by (1), 

d(l0gZ) a (log 3) , 

dz mdx 

z 

P 
we obtain 



substituting for m its value — , and separating the variables, 



(llOfCZ) d(\ogx) / 3 x 

dz dx v ' 

This equation is true for all values of x, z } dx, and dz, and may 
be written in the form 

zf (*)=**/'(*) ; ( 8 ') 

moreover, the values of x and z are entirely independent, be- 
cause their assumed ratio m has been eliminated. Now, since 
the form of the first member of equation (3) or (3') is the 
same as that of the second member, this equation shows that 
the value of the expression x ~^^~ is unchanged when x is 
changed to z, the latter denoting any other value of the inde- 
pendent variable, that is, the expression is independent of the 



THE DIFFERENTIAL CALCULUS. 23 

value of x \ it will be found, however, to depend upon the base 
of the system of logarithms to which log# belongs. 
Denoting now the base by b, we put 

X ^A =B (4; 

In this equation, B can be dependent upon no quantity ex- 
cept 6. Equation (4) may be written in the form 

dOeg b x) = ^ (5) 

similarly, if a is the base, we use A to denote the constant 
value of the expression in equation (3), thus — 

d(h&x) = -^L (6) 

A relation between A and B is found by differentiating, by 
(5) and (6), the identical equation 

log a s = log a 6 log b z,* (7) 

and thus obtaining, 

Adx , . Bdx 
__=log a &-_; 

whence 

A=Blo S& b=hgJ»; 
therefore, by the definition of a logarithm, 

a k = b* (8) 

Now, it is obvious that the value of a K cannot depend upon 

* This identity is mogt readily obtained thus, — by definition 

taking tho logarithm to the base a of oaoh member, we have 
log*.? SB \0g\jX logafc. 



24 PROPOSED METHOD OF TREATING 

b 7 hence equation ( 8) shows that the value of b B likewise can- 
not depend upon b ; b B must, therefore, have a value entirely 
independent of the base of the system of logarithms to which 
log# belongs. Denoting this constant value by e, we write 

V=e (9) 

Taking the logarithm of each member of equation (9), adopt- 
ing this constant as a base, we have 

J51og e 5=l, 

"Whence 

*-TS* (10) 

Introducing this value of B in equation (5), we obtain 

dx 



c?(log b z): 



logeb.X ' 

If the logarithms are taken in the system whose base is e, 
the last equation reduces to 

d(log.*)«-f • (11) 

19. The constant e is the base of the Napierian system of 
logarithms. In Art. 22, it is shown that this constant is 
greater than 2, and less than 3 : the more exact determination 
of its value is however deferred until the student is able to de- 
duce the requisite series by means of Maclaurin's Theorem. 

Articles 20, and 21, upon which Art. 22 depends, will serve 
incidentally to show how the method of rates is applied to 
finding the formula for the differential of an area, 

20. The equation of the hyperbola referred to its asymptotes 
is 



THE DXFEEBENTIAL CALCULUS. 



25 



taking c^=l, and the axes rectangular, we have for the equa- 
tion of the rectangular hyperbola in the figure, 
1 1 

Draw the ordinate of the vertex (JL, 1), and suppose a vari- 
able ordinate to move from this position toward the right. The 
area generated by this ordinate will be an increasing variable 
quantity, which may be regarded as a function of x i and de- 
noted by F{x). 




B R 

At the instant x has attained a given value, the correspond- 
ing rate of F(x) may be measured by giving the area a con- 
tinuously uniform rate identical with that which it has at that 
instant [see Art. 8]. This is effected by supposing y to re- 
main constant, and x to increase uniformly from and after the 
given instant with the rate dx, thus rendering the rate of the 
area uniform. In the figure, the area A B R P represents the 
value of F(x) at the given instant, and the rectangle ydx its 
rate at the same instant. Hence wo have at every instant 
d{ F{x) ] =tf(Aroa)=ycfo. 



26 PROPOSED METHOD OF TREATING 

This equation, it may be remarked, has been derived without 
regard to the relation between y and x } and is therefore appli- 
cable, whatever be the equation of the curve. 

21. In the present case, since y = , we have 

by equation (11) of Art. 18. Thus F(x) y the expression for the 
area, varies at the same rate as the Napierian logarithm of x. 

Now, when # = 1, the area reduces to zero, likewise log e l = 0, 
or 

therefore, F(x) and log e r, starting from the same value, and con- 
stantly increasing at the same rate, we have 

F(x)=log,x. 

That is, the area included between the hyperbola, the axis of 
x, and the ordinates corresponding to x and 1 respectively, is 
equal to the Napierian logarithm of x. It is easily inferred by 
subtraction that the area included between any two ordinates 
is the logarithm of the ratio of the abscissas. 

22. Let the ordinates corresponding to the abscissas 2 and 
3 be drawn ; then the area included between the ordinates cor- 
responding to 1 and 2 will represent log e 2, and that included 
between those corresponding to 1 and 3 will represent log e 3. 
Completing the square AD2 1, we see that log e 2 is less than 
one. 

Draw a tangent to the curve at the point (2, i) ; this tan- 



THE DIFFERENTIAL CALCULUS. 



27 



gent, the axis of x, and the ordinates corresponding to 1 and 
3 form a trapezoid of which the sum of the parallel sides is 




1 2 3 

one, and the distance between them two ; its area is there- 
fore one, and the area representing log e 3 is greater than one. 
Therefore log e 3 > 1 > log e 2. 

Now, since log e c=l, this inequality is equivalent to 
log e 3 > log 6 > log,2 
or 3 > e > 2. 

The Napierian base is therefore intermediate in value be- 
tween 2 and 3. 

The differentials of exponential functions of the forms a x and 
x y are easily derived from the formula for differentiating the 
logarithmic function. 



THE DIFFERENTIALS OF THE SINE AND THE COSINE. 

23. Let the variable angle bo generated by the rotation of. 
the radius a about the origin of rectangular co-ordinates start- 



28 



PROPOSED METHOD OF TREATING 



The extremity P of the radius 

(i) 



ing from the position OA. 

moves in the circle, 

x> + y 1 ~— a 2 

and generates the variable arc s. 

Let PB and BP represent the rates respectively of the 
abscissa and ordinate of P when it arrives at the position in- 
dicated in the figure. Then will PP be a tangent as in Art. 12; 
and, moreover, the length PP' will represent the actual velocity 

kP 1 




ds, of the point P, because it is the space through which P 
would move in the unit of time, were dx, dy } and consequently 
ds> to become constant. 

Now, sin0=-~-, and cosd = ~ , (2) 



dy 



a ' 
dx 



d(smO) = 2L , and d(cosd) = — (3) 

In equations (3) we have to express dy and dx in terms of 
6 and dO. 



THE DIFFERENTIAL CALCULUS, 29 

Denoting by <p the inclination to the axis of x of the direc- 
tion of the motion of the generating point when s is increasing, 
we have 

dy = Bin^.ds, and dx = cos(j>.ds } (4) 

and, substituting in equations (3), we obtain 

ds ds 
d(&m)d = sin</>. — , and <#(cos#)= cos</>. — (5) 

In the figure, (p being in the second quadrant, and ds being 
positive, dx is negative. 

Differentiating equation (1), 

%dx-\-ydy=zQ, 



whence 



to *=sr MT 5 r. ( 6 ) 



=tan0, , (7) 



or, since 

1 

x 

tan^— — cot#=tan(#4:^7r); 
therefore 

0=0+4* (8) 

In equation (8) we take 0+|tt, because has been defined 
as the angle between the positive directions of ds and dx. 
Now, in equations (5), we substitute 

sin0 = sin(# + i7r)=costf, 

cos0 =cos(#-h \tt) — — sin#, 
and 

= d$, since - =0. 
a a 



30 PEOP08ED METHOD OF TREATING 

Whence 

d r (sin<9) = cos Odd, 
and 

d(co80)=-sin0tf0. 

24. Equation (8) shows that, in the case of the circle, the tan- 
gent, according to the general definition of Art. 12, is perpen- 
dicular to the radius through the point of contact, and is there- 
fore identical with the tangent as defined in Geometry. 

The differentials of the remaining circular functions may be 
deduced from the above formulas in the usual manner. 



25. In conclusion, we will briefly indicate the mode of 
adapting this method to applications of the Calculus. 

The rate method is, by its definitions, peculiarly adapted 
to Kinematical and Physical Investigations. (See extract 
from Coumot, Art. 7 of this paper). It is noticeable that 
in several of the best recent treatises on Dynamics, Newton's 
Fluxional notation as applied to functions of the time, has been 
revived, and is freely employed. 

26. In the Integral Calculus, as well as in the Differential, 
the rate method excels in the clearness it gives to the funda- 
mental definitions. 

The Indefinite Integral is defined as such a function of the 
independent variable as will vary with the rate expressed 
under the integral sign. A Particular Integral is any one 
such function. It is then easily shown that the particular 



THE DIFFERENTIAL CALCULUS. 31 

integrals included in an indefinite integral can differ only by 
a constant. A Definite Integral is the amount by which a 
function, having the rate expressed under the integral sign, 
actually changes, while the independent variable passes from 
one limit to the other. Accordingly, an indefinite integral 
becomes " particular ,f when the lower limit is fixed, the upper 
limit remaining variable. 

27. In deriving differential expressions to be integrated, 
a reference to the pages of Mo?ilucla, cited in the introduction 
to this paper, Art. 4, will show that the methods employed 
in works on Fluxions, for deducing the formulas for the rates 
of areas* and volumes were in general both clear and satisfac- 
tory. The underlying principle is that of rendering a non- 
uniform rate measurable by removing the cause of its variable 
character (in this case, the cause is the variation of the 
generating line or area). 

In deriving other formulas, like those for moments and 
pressures, the methods employed in these treatises were not 
so satisfactory, and at this point we propose to show that the 
comparative rates of two quantities can he obtained by taking 
the ratio of simultaneous increments and passing to the 
limit. 

It is thought that the introduction of limits at this stage of 
his progress will present little difficulty to the student, now 
familiar with the notion of continuity ; while it will prepare 

* Sec also Article 20 of this paper. 



32 THE DIFFERENTIAL CALCULUS. 

him for the study of writers who employ limits and infin- 
itesimals in the applications of the Calculus; and serve to 
demonstrate the fundamental identity of the different methods 
of treating this important branch of mathematics. 



Price 50 Gents. 



ON 

A NEW METHOD 



OF OBTAINING THE 



Differentials of Functions 

WITH ESPECIAL REFERENCE TO THE 

NEWTONIAN CONCEPTION OF RATES 
OR VELOCITIES 

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PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY 
AND 

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PROFESSOR OF MATHEMATICS IN SAINT JOHN'S COLLEGE ANNAPOLIS MARYLAND 



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